# Price elasticity of demand always increases with price?

Is there a closed-form continuous demand function whose price elasticity of demand decreases with the price?

Taking hint from 1muflon1's answer, consider the following demand function:

\begin{align} Q &= e^{1/p} \\ \frac{dQ}{dp} &= -\frac{e^{1/p}}{p^2} \\ \frac{dQ}{dp}\frac{p}{Q} &= -1/p \end{align}

So absolute elasticity, $$1/p$$ is decreasing.

There are many functions where absolute value of elasticity is decreasing and which are continuous closed form.

One example of such function would be:

$$Q= a-\ln(p), p\geq1 \implies EL = - \frac{1}{p}\frac{p}{a-\ln p} = -\frac{1}{a - \ln p},$$

which is continuous closed form and decreasing in price in its absolute value.

Originally I interpreted the request too mathematically I am leaving it here since other answer references it.

Yes for example consider the following demand function:

$$Q = e^{-p}$$

price elasticity of demand is given by:

$$EL = \frac{dQ}{dp}\frac{p}{Q} = -e^{-p} \frac{p}{e^{-p}} = -p$$

So the demand function has elasticity that is always decreasing in price $$p$$. The function has closed form, and it is continuous.

• I should have made it clearer. I was talking about the absolute value of elasticity. – Adam Nov 30 '20 at 1:44
• @Adam yes I interpreted your request too mathematically anyway I gave you additional example – 1muflon1 Nov 30 '20 at 8:23
• @1muflon1: I think you don't need the negative sign. $Q=\ln p$ is what you perhaps meant. – Dayne Nov 30 '20 at 9:51
• @Dayne we need negative sign just so the demand function makes sense $Q=\ln p$ would make sense only for giffen goods and even there with a caveat as giffen goods are often giffen goods only for a portion of demand curve. (you can try to plot $\ln p$ and $-\ln p$, you will see what I mean. I mean we should also probably add some constant to raise the demand above zero I will add that – 1muflon1 Nov 30 '20 at 9:56
• The absolute value of the elasticity would be 1/Q, increasing in p. Dayne's works. – Adam Nov 30 '20 at 20:22

In theory, demand should decrease when the price increases. But in practice, there are other psychological factors which might result in far less predictable behavior of elasticity of demand.

Sometimes consumers will assume that a product with a low price is of inferior quality (despite absence of any other indicators that this is the case), and will instead choose to buy a more expensive substitute good. So lowering the price can counter-intuitively result in a reduction of demand.

There are even some rare cases of product where the price elasticity of demand appears to be reversed: The more expensive it gets, the higher the demand. This is the case for luxury goods people buy for no other reason than to serve as status symbols to show off their wealth. You can't demonstrate wealth with things you got for cheap. So there might be a higher demand for 10,000 € gold watches than for practically identical 1,000 € gold watches... as long as it is easy to tell them apart.